Subjects
A cohort of 10 healthy volunteers aged between 21 and 38 were recruited. Ethical approval was granted by National Health Service Health Research Authority NRES Committee Southeast Coast-Surrey with reference number 06/Q0602/18. The image acquisition protocol and the translation of magnet resonance images (MRI) to 3D virtual airway surfaces were the same as in previously published work. 20
The ethical approval was granted by the National Health Service Health Research Authority NRES Committee South East Coast-Surrey with reference number 06/Q0602/18. Informed consent was obtained from each of the subjects included in this study. All methods were carried out in accordance with relevant guidelines and regulations.
CFD Simulations and Boundary Conditions
1. CFD simulations
The CFD software STAR-CCM+ (Siemens PLM Software, Plano, TX) was used to model the heat and water transported by the air flow through the nasal cavity. The flow computations were performed for inspiratory flowrates at 15 and 30 \(L.mi{n}^{-1}\) using the methodology as described more fully in previous work; briefly a mass flow inlet was specified at a far-removed external domain inlet and a pressure outlet condition was specified at an extended nasopharyngeal outlet. 20 There are 4 million total CFD polyhedral volume cells for each CFD simulation, and each simulation has 7 prism layers near the airway wall to account for both high flow and temperature gradient. The total thickness of prism layer is 0.3 mm, and first layer is 0.01 mm with a stretch factor of 1.7 to allow a smooth transition of cells in between. More detailed mesh scenes can be found in the previous study on nasal airflow distribution in this cohort.20 Results using our model for heat and water transfer are compared with previous in-vivo measurements in the results section.
Heat and water vapor boundary conditions
The transport of heat and water vapor by the airflow may be described respectively by:
\(\frac{\partial T}{\partial t}+ \left(\text{u}\cdot \nabla \right)T=\frac{k}{{\rho }{C}_{p}}{\nabla }^{2}T \left(1\right)\) \(\frac{\partial {\text{Y}}_{{\omega }\text{v}}}{\partial \text{t}}+\left(\text{u}\cdot \nabla \right){Y}_{{\omega }\text{v}}=D{\nabla }^{2}{Y}_{{\omega }\text{v}} \left(2\right)\)
where \(T \left(K\right)\) is temperature, \(u (m/s)\) is the flow velocity, \(k (W\cdot {m}^{-1}\cdot {K}^{-1})\) thermal conductivity, \(\rho (kg\cdot {m}^{-3})\) density, \({C}_{P} (J\cdot k{g}^{-1}\cdot K)\) specific heat capacity, \({\nabla }^{2}\) is the Laplace operator, \({Y}_{\omega v}\) the water vapor fraction and \(D (c{m}^{-2}\cdot {s}^{-1})\) the diffusion coefficient of water vapor in air. At the mucosal wall, a two-film model was employed to describe the exchange of heat and water with the mucosal surface.
Figure 1 shows schematically the modelling of heat and water vapor transfer from the nasal cavity wall into the inhaled air. The layer labeled “Organ” represents the nasal capillary bed, “Membrane” represents the layer of liquid mucus and the upper portion of the water supplying mucosal tissue on the nasal wall; the first wall layers represent the air as discretized in the CFD mesh. The organ layer is responsible for providing the heat source in the model. This heat must then flow through the membrane to the air. The membrane layer is assumed to have the same specific heat capacity as water.21 The temperature at the interface (\({T}_{s}\)) between the exposed surface of the nasal wall and the cavity air is determined by the resultant heat flux. The inhaled air temperature is represented by \({T}_{a}\), whilst \({T}_{o}\) refers to the temperature of the capillary bed. The species boundary condition at the interface is assumed to be fully saturated at the local surface temperature.
The total amount of heat flux into the air (\(q{̇}_{air}\)) is the sum of the flux from the membrane (\(q{̇}_{membrane}\)) and the latent heat flux (\(q{̇}_{latent}\)) carried by the water during evaporation (or condensation):
$$q{̇}_{air}=q{̇}_{latent}+q{̇}_{membrane}$$
The fluxes \(q{̇}_{air}\), \(q{̇}_{latent}\) and \(q{̇}_{memb}\) are calculated using the equations below:
$$q{̇}_{air}=-{k}_{air}\frac{{T}_{\text{a}}-{T}_{s}}{{{\delta }}_{\text{w}\text{a}\text{l}\text{l}}}$$
$$q{̇}_{latent}={L}_{{H}_{2}O}\cdot \dot{{\omega }}$$
$$q{̇}_{membrane}=-{k}_{membrane}\frac{{T}_{\text{s}}-{T}_{o}}{{{\delta }}_{\text{m}\text{e}\text{m}\text{b}\text{r}\text{a}\text{n}\text{e}}}$$
where \(\dot{{\omega }} (kg\cdot {m}^{-2}\cdot {s}^{-1})\) is the water mass flux between the fully saturated wall and the cavity air, and \({{\delta }}_{wall} \left(m\right)\) is the half thickness of the first wall layer of the CFD mesh as labelled in Fig. 1. The latent heat of water \({L}_{{H}_{2}O} (J\cdot {g}^{-1})\) is defined by the formula: 22
$${L}_{{H}_{2}O}=2500.8-0.00006\cdot {T}_{s}^{3}+0.0016\cdot {T}_{s}^{2}-2.36\cdot {T}_{s}$$
,
where \({T}_{s}\) is in Celsius. Using Fick’s law of diffusion, \(\dot{{\omega }}\) is calculated as:
$$\dot{\omega }= -{\rho }_{fluid}\cdot D\cdot \frac{\partial {Y}_{wv}}{{\partial \delta }_{wall}}$$
where \({\rho }_{fluid}\) is the density of the mixture of air and water vapor. The mass fraction of water vapor under fully saturated conditions was quantified from an empirical fit of psychrometric data by the formula below. 23
$${Y}_{wv}=2\cdot 1{0}^{-5}{T}_{s}^{2}+0.0003{T}_{s}+0.0025$$
Heat flux and water flux into the air were determined by the surface temperature \({T}_{s}\), and were iteratively updated so that heat and water fluxes at the membrane and at the air were balanced at each time step. Combining the above equations, the surface temperate \({T}_{s}\) can be calculated as:
$${T}_{s}=\frac{{k}_{air}{T}_{a}+{k}_{membrane}{T}_{o}-{L}_{H2O}\dot{{\omega }}{{\delta }}_{wall}{{\delta }}_{membrane}}{{k}_{membrane}{{\delta }}_{wall}+{k}_{air}{{\delta }}_{membrane}}$$
For this cohort study, computations were performed with \({T}_{a}\) set to 25\(^\circ C\), 50% relative humidity (corresponding to a mass fraction of 0.01125 and \({T}_{o}\) = 34\(^\circ C\)). The rendering of the applied both temperature and moisture boundary conditions can be found in appendix.
3. Geometric definition
The segmented 3D surface from the MRI scan of each subject includes anatomy from the face to the end of the nasopharynx. In addition, for each subject, an external half sphere (diameter = 0.5m) was attached to the face to ensure natural inflow profile and an extrusion (order of 50 diameter) was added from the end of nasopharynx to prevent reverse flow. Figure 2 illustrates the reconstructed 3D surface of one subject with the anatomical landmarks used, as defined in our previous flow only study. 20
Measurements
1. Air Temperature
The mean air temperature was calculated over each of the cross-sectional plane (Fig. 2 right) from anterior to posterior cavity in each subject; in addition, the cohort mean air temperature at each corresponding plane was also calculated.
2. Relative humidity and moisture content
Relative humidity (RH) is the ratio of the mass fraction of water vapor mixture to the mass fraction of water vapor if fully saturated, at the corresponding temperature and pressure. Lindemann et al., reported that RH can be 90% at 32 \(^\circ C\) in the nasopharynx, however, RH is a temperature and pressure dependent metric, and it is not ideal for inter-subject comparisons, given the absolute amount of water vapor is the metric of interest. Therefore, moisture concentration was compared to alveolar conditions (100% RH at 37 \(\text{C}\)), rather than local conditions, using a new quantity defined as the ratio between mass fraction of H2O at local conditions and the mass fraction of H2O at alveolar conditions, which was named moisture content (MC). The equation below shows the calculation of MC where \({\text{Y}}_{t}\) is the local mass fraction of water vapor, while \({\text{Y}}_{37℃}\) refers to the mass fraction of water vapor at alveolar conditions.
$$MC=100\cdot \frac{{\text{Y}}_{t}}{{\text{Y}}_{37℃}}$$
Thus, MC is suitable to quantify how much the water vapor is in the air compared to alveolar conditions, irrespective of local conditions.
Intra and inter-subject variations in MC were plotted in a similar manner to the temperature variations described above.
Film coefficients
To quantify the efficiency of heat and water transfer between the nasal cavity wall and inhaled air, heat and mass film coefficients were introduced. The equation below shows the calculation of the heat film coefficient. The left side of the equation \(Q \left(W\right)\) represents the heat flux while the right side of the equation is the product of heat film coefficient \(h (W\cdot {m}^{2}\cdot K)\), nasal cavity surface area \(A \left({m}^{2}\right)\) and temperature gradient at the wall (\({\Delta }T\)):
$$Q=h\cdot A\cdot {\Delta }T$$
Analogously to heat transfer, the mass film coefficient was calculated using the formula below, indicating water vapor flux is a product of mass film coefficient \({h}_{c} (m\cdot {s}^{-1})\), nasal cavity surface area \(A \left({m}^{2}\right)\) and gradient of mass fraction of water vapor at the wall (\({\Delta }{c}_{A}\)):
$${{\Omega }}_{A}{=h}_{c}A{\Delta }{c}_{A}$$
Data analysis
The Wilcoxon signed-rank test was performed to evaluate the statistical significance of decongestion-associated changes in nasal cavity air-conditioning efficiency.24 Following convention, a 95% confidence level was chosen to distinguish significant from non-significant results. 25 Box and whisker plots were used to illustrate the results.